The present invention relates generally to three-dimensional (3D) computerized tomography (CT) and more particularly to the generation of Radon derivative data on a near uniformly spaced polar grid in a digitized Radon space.
In conventional CT for both medical and industrial applications, an x-ray fan beam and a linear array detector are used to achieve two-dimensional (2D) imaging. While the data set may be complete and image quality is correspondingly high, only a single slice of an object can be imaged at a time. Therefore, when a 3D image is required, a stack of 2D slices approach is employed. Acquiring a 3D data set, one 2D slice at a time is inherently slow. Moreover, in medical applications, motion artifacts occur because adjacent slices are not imaged simultaneously. Also, dose utilization is less than optimal because the distance between slices is typically less than the x-ray collimator aperture, resulting in double exposure to many parts of the body.
In order to overcome the problems associated with the x-ray fan beam and linear array detector configuration, a cone beam x-ray source and a 2D array detector are used. With the cone beam x-ray source and linear array detector, the scanning is much faster than the slice-by-slice scanning of the fan beam. Also, since each "point" in the object is viewed by the x-rays in 3D rather than in 2D, a much higher contrast can be achieved than is possible with the conventional 2D x-ray CT. To acquire cone beam projection data in the cone-beam CT implementation, an object is scanned, preferably over a 360.degree. angular range, either by moving the cone beam x-ray source in a scanning circle about the object, while keeping the 2D array detector fixed with reference to the cone beam x-ray source or by rotating the object while the x-ray source and detector remain stationary. The image of the object can be reconstructed by using a Radon inversion process, in which the total Radon transform of the cone beam projection data is computed. Computing the total Radon transform in the continuum Radon space requires a large amount of processing power and an infinite amount of time. In order to overcome this computing problem, the cone beam projection data is sampled so that the Radon transform is computed for a finite set of uniformly spaced points. The best results occur by sampling the Radon space into a polar grid having a plurality of uniformly spaced grid points. In a discrete Radon space, it is desirable to compute Radon derivative data on each of the uniformly spaced grid points. However, in order to compute Radon derivative data on each of the uniformly spaced grid points, there has to be a continuum of source positions located on the scanning trajectory. In practice, there is only a finite number of source positions and not a continuum of source positions. Thus, Radon derivative data cannot be precisely computed on each of the uniformly spaced grid points.